7. The turnbuckle T is tightened until the tension in cable OA is 5 kN.

Express the force acting on point O as a vector. Determine the projection

of onto the y-axis and onto line OB. Note that OB and OC lies in the x-y

plane.

F

F

8. Determine the magnitude and

direction angles of the resultant force

acting on the bracket.

kjiF

xyxyFF

45sin45030cos45cos45030sin45cos450

11

1

21 FFFR

kjiF

2.31857.2751.1591

Resultant

kjiF

120cos60060cos60045cos6002

kjiF

30030026.4242

Direction angles for 𝑭 𝟐 906045 zyx

1coscoscos 222 zyx

1222 nml

Direction cosines

1cos60cos45cos 222 z

5.0cos25.0cos2 zz

1205.0cos90 zzz

kjiF

30030026.4242

Direction angles for 𝑹

029.097.633

2.18cos907.0

97.633

57.575cos418.0

97.633

16.265cos

coscoscos

zyx

zz

y

yx

xR

R

R

R

R

R

Direction Cosines of Resultant Force

3.88)029.0arccos(9.24)907.0arccos(3.65)418.0arccos( zyx

kjiF

2.31857.2751.1591

Resultant 21 FFFR

kjiR

3002.31830057.27526.4241.159

kjiR

2.1857.57516.265

Magnitude of Resultant Force

NRR 97.6332.1857.57516.265 222

9. Determine the parallel and normal components of force 𝐹 in vector form with

respect to a line passing through points A and B.

35

50°x

y

z

A

B

(3, 2, 5) m

(6, 4, 8) m

F=75 kN

F

kNFxy 311.6435

575

22

kNFF xyx 34.4150cos

kNFF xyy 265.4950sin

kNFz 59.3835

375

22

kjiF

59.38265.4934.41

Cartesian components of 𝑭

35

50°x

y

z

A

B

(3, 2, 5) m

(6, 4, 8) m

F=75 kN

F

kjiF

59.38265.4934.41

Unit vector of line AB

kjikji

nAB

639.0426.0639.0

323

323

222

Parallel component of 𝑭 to line AB (its scalar value) :

kNF

kjikjiF

nFF AB

14.72639.059.38426.0265.49639.034.41

639.0426.0639.059.38265.4934.41

//

//

//

Parallel component of 𝑭 to line AB (in vector form):

kjikjinFF AB

14.4676.3014.46639.0426.0639.014.72////

Normal component of 𝑭 to line AB (in vector form):

kjiFFF

55.7505.188.4//

ABn

10. An overhead crane is used to reposition the boxcar within a

railroad car-repair shop. If the boxcar begins to move along the rails

when the x-component of the cable tension reaches 3 kN, calculate

the necessary tension T in the cable. Determine the angle xy between

the cable and the vertical x-y plane.

Tx=3 kN, calculate tension T, the angle xy between the cable and the vertical x-y plane.

x-component of 𝑻

kjin

kjin

T

T

154.0617.077.0

145

45

222

kjiTT

154.0617.077.0

kNTT 896.3377.0 (magnitude of 𝑻)

kNTkNT zy 6.0896.3154.04.2896.3617.0

y and z-components of 𝑻

kjiT

6.04.23

Unit vector of 𝑻

𝑻 in vector form

kNTTT yxxy 84.34.23 2222

598.9896.089.3

84.3cos xy

xy

xyT

T

11. The spring of constant k = 2.6 kN/m is attached to the disk at point A and

to the end fitting at point B as shown. The spring is unstretched when A and

B are both zero. If the disk is rotated 15° clockwise and the end fitting is

rotated 30 ° counterclockwise, determine a vector expression for the force

which the spring exerts at point A.

12. The rectangular plate is supported by hinges along its side BC and

by the cable AE. If the cable tension is 300 N, determine the projection

onto line BC of the force exerted on the plate by the cable. Note that E is

the midpoint of the horizontal upper edge of the structural support.

If T=300 N, determine the projection onto line BC of the force exerted on the plate by the cable.

x

y

z

𝑪𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔 𝒐𝒇 𝒑𝒐𝒊𝒏𝒕𝒔 𝑨,𝑩, 𝑪 𝒂𝒏𝒅 𝑬 𝒘𝒊𝒕𝒉 𝒓𝒆𝒔𝒑𝒆𝒄𝒕 𝒕𝒐 𝒄𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆 𝒔𝒚𝒔𝒕𝒆𝒎

𝑨 400, 0, 0 B (0, 0, 0)

𝑪 0, 1200𝑠𝑖𝑛25,−1200𝑐𝑜𝑠25C (0, 507.14, 1087.57)

𝑬 0, 1200𝑠𝑖𝑛25,−600𝑐𝑜𝑠25E (0, 507.14, 543.78)

kjiT

kjinTT T

21.19319.18012.142

33.844

78.54314.507400300

kjkjinTT BCBC

906.0423.021.19319.18012.142

Unit vector of 𝒍𝒊𝒏𝒆 𝑩𝑪

z

y

25o25o

BCn

kjjknBC

906.0423.025sin25cos

Projection of T onto line BC

NTBC 26.251

13. The y and z scalar components of a force are 100 N and 200 N,

respectively. If the direction cosine l=cosx of the line of action of the

force is 0.5, write 𝐹 as a vector.

5.0200100 lNFNF zy

75.05.011 22222222 nmnmnml

yzzy FNFF 61.22322

FnFFFmFF zzyy coscos

NFF 2.25861.22375.0

NFF xx 1.129cos

kjiF

2001001.129